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Time to Hire a Monkey? Not Really

Recently I have noticed a renewed interest in a few blogs and forums about “monkey style trading”. This type of trading usually involves coin tosses, percent stop-losses for risk management and trailing stops for profit taking. This is how it works in principle, according to its proponents (which happen to be humans by the way, not monkeys):

(1) A coin is tossed to decide whether to go long or short.
(2) After a position is opened a stop is established as a percent of bankroll, usually 1% – 2%
(3) A trailing stop is set to capture the price trend
(4) When a position is closed a new one is opened by tossing again the coin

There are several variations of steps 2 and 3 in terms of how position size is determined and the trailing stop is adjusted. I will not go into these details because they are boring.

An important first thing to realize about this style of trading is that backtest results will vary depending on initial conditions, i.e. whether at the beginning of the price history there is an uptrend, downtrend or a sideways market in relation to the outcome of the first toss.

Another more important realization is that depending on tossing outcome sequences, one may “toss” himself to ruin by constantly hitting a stop-loss. Of course, the probability for that is very low and decreases as the stop-loss decreases but it is always finite, as explained in detail in my book “Profitability and Systematic Trading”. As a result, “monkey style” trading systems differ significantly from systems with deterministic entries in that if you have two traders using the same system, one may win and the other may lose, meaning that results are random because they depend on coin tossing sequences in relation to market trend and volatility.

An example

Suppose that we are testing an instrument with price history that includes just an upward smooth trend (no volatility), for illustration purposes only. If the outcome of the first coin toss is a long position then this will be a 100% profitable system with just one trade and profit equal to 1 (for simplicity). If the outcome is a short position, then there will be an initial loss and then a new toss. Let us assume for the shake of this simple example that the loss is 1/2 of the price trend. If the next toss is a short position then the system will be a 100% loser with 2 trades. If the next toss is a long then the system will be 50% profitable with 0 profit (we assume negligible commissions for simplicity). The expectancy of this system is found by adding all the expected rewards multiplied by their probability as follows:

E = 0.5 × 1 + 0.5 × (-0.5) = 0.25

Some newbies jump right away when they calculate a positive expectancy as in this case because they think they found the holy grail. It is not always so and I will attempt to explain where the fallacy lies in this case. It is a subtle fallacy, an understanding of which separates quantitative analysis from wishful thinking and alchemy.

The expectancy of a trading system is the amount it wins on average in the sense of the law of large numbers. In the context of the simple trading systems example above, it is the amount won on average when the number of trades, and as a result of tosses, becomes very large. In the case of a typical deterministic trading system, we all understand what it means having a large number of trades: it means having a sufficiently large sample to guarantee statistical significance of the expectancy number. But in the case of “monkey style” trading systems, this also presupposes having a large number of tosses. But how large? In the example I gave you only need two tosses to finish the game. Two tosses is not enough for statistical significance. The expectancy will be statistically significant only if this game is played many times, something that makes no sense in the case of trading systems because a trader has only one chance to trade a market. It will be a significant result only if you consider a large number of traders playing the game simultaneously (or a large number of instruments as it will be discussed later). Anyone with basic knowledge of probability theory understands that if there is a number of traders N playing this game, as N becomes large, then – for this example only – N/2 traders will win right from the first toss since the probability of long equals the probability of short, both being equal to 1/2. Out of the remaining 50% of the traders, 25 % will turn losers and 25% will break even after the second toss. Thus, this system with a positive expectancy is only profitable for 50% of the traders.

Therefore, this is not the expectancy of a trading system but the average, or expected gain, of either a large pool of traders that all play it once, or of a trader that plays it a large number of times. Thus, this is not the expectancy of a specific trading system but an artifact of the law of the large numbers.

What happens if the stop-loss is reduced to 1/4 instead of 1/2 in the simple example above? One can show that on the average, 56.25% of the traders will win. Only when the stop-loss is near zero, the number will approach 100% asymptotically. But as the stop-loss becomes small, profitability is seriously reduced by commissions. If you add volatility to that, the expectancy may even get negative.

Hire many monkeys instead?

Some have proposed trading many instruments with this system in order to get statistical significance. For example, one could apply this system to a universe of stocks, like the constituents of S&P 500, or to many commodities futures contracts. In this case, the number of trades is large and the number of tosses is large to guarantee sufficient samples. However, one must consider the effect of commissions in this case and the starting capital required to trade such a random system with many instruments. Trading cost escalates and may result in complete exhaustion of even very large starting accounts depending on instrument correlations. In the case of one instrument, I have already demonstrated how commission cost can result in total ruin very fast although studies that did not include commissions showed spectacular performance. Again, an artifact of neglecting a very significant wealth-out effect in the markets, commissions.

Make monkeys a little more intelligent?

A variation of “monkey style” trading involves reversing direction after the first random trade that hits a stop-loss and taking a random trade again only after a winner. Obviously, this system involves some rudimentary timing technique and it is not pure “monkey style” trading. Nevertheless, this type of system suffers seriously during sideways markets unless the stop-loss is set too large, which in turn causes large drawdown when an opposite trend develops with respect to position taken.

There is no free lunch in the markets. Profitable trading requires good timing of both entries and exits. This is a fact experienced traders have come to know, some the hard way. Systems with random entries that appear to have positive expectancy are mostly the artifacts of the law of large numbers and have no practical use because they often do not tell us the percentage of traders that are losers but only average performance of a large number of traders. No serious trader, especially a fund manager, would use systems that do not have explicit and well-defined rules for establishing positions. I can’t imagine anyone managing 10 billion dollars, for example, tossing a coin in order to take a position, although it is a fact that many deterministic systems do worse than coin tosses.

Have a great weekend!