CHAPTER 1
Martingale and Anti-Martingale

1.1 The Right Stake

As mentioned in the introduction, money management (M.m) aims to establish the best stake to place when opening a trade or, in general, how much of your capital to use in the gamble you are about to embark on.

I think we all tend to adopt quite a simple statistical approach that encourages us to hope in a positive result after one or more negative results, and to fear repeating a success after placing a successful stake. In general, this is why you don't want to continue after a certain number of consecutive winning trades, while after a series of losing trades you'll be sure the next one will be a winner.

This tendency induces us to adopt a sort of risk management that, in general, leads us to increase the stakes after a negative period (betting on the fact that after various losses one should statistically expect a success) and reduce them after a positive period (for exactly the opposite reason).

In this chapter, we'll deal with this question by moving away from the trading environment, to enter a world we're all in any case familiar with: that of the coin toss.

Flipping a coin to see whether it lands heads or tails is a classic statistical example of 50% probability, and analyzing how we manage the stakes, on the basis of one event or another, can produce some surprising results.

This isn't trading, and the intention isn't to compare a trading system to betting. The purpose of this first part is simply to demonstrate what might be the best way to manage your available capital, when 'staking' part of it on an event.

If we take 100 people with ?100 each, I don't think many would come out winning if they had to bet on a series of 100 or 1,000 coin tosses. In my opinion, most would lose all their capital due to inadequate risk management.

Of the resources to download, at the link https://autc.pro/guide you'll find the Excel file 'HeadOrTail.xls' you can use to run coin-toss simulations. This is the one I used for the various examples we'll be taking a look at.

As I said, let's suppose we have a capital of ?100 and we'll use it for a series of 100 and 1,000 coin tosses, 'heads' wins, 'tails' loses. The win/loss ratio will be different for each analysis. In other words, let's imagine we lose ?1 on every stake; the amount won, on the other hand, changes as we analyze various examples.

Stake calculation systems are mostly based on two styles that can be grouped together as Martingale systems and anti-Martingale systems. The first aim to increase exposure in the case of a loss; while the second only increase exposure after a win and decrease it in the case of a loss.

1.2 Martingale

The Martingale system comes from the roulette wheel, and in practice is based on the impossibility of an infinite series of consecutive losses. Therefore, the concept is that the more consecutive losses there are, the greater the probability of a win next time. On this basis, the system involves doubling the stake after every loss. If you bet 1 on the first spin of the wheel, you'll bet 2 on the second if the first bet lost, and if you lose again you'll bet 4, then 8, and so on, and when you get a winning spin of the wheel you'll finally have made a profit. Note that, if you get a win on the second spin, you'd win 2, and after losing 1 on the first spin you'd be 1 up. If you lost also on the second spin, you'd have lost 1 + 2 = 3, so winning 4 on the third spin would again give you a profit of 1. If you lost on the third spin, you'd have lost 1 + 2 + 4 = 7, and winning on the fourth spin would make 8, giving you a profit of 8 - 7 = 1. As this simulation continues, we can see that, when we finally win, we make a profit of 1, just like we would have if we'd won on the first bet.

The above is true if you double the stake, and it's closely related to roulette-betting systems where one bets on red and black or odd and even numbers. In much more general terms, all approaches that simply increase the stake after a loss, and not just ones that double it, are called Martingale approaches; vice versa, these approaches decrease the stake after a win.

I'd like to emphasize that most people probably have a natural inclination to prefer a Martingale-type approach.

Now let's take a look at the simulations. The first is based on the supposition that, a win produces a profit of ?1.25 for each ?1 bet, while a loss loses the ?1 bet. As mentioned above, the probability a coin toss comes down heads is 50%, so out of 1,000 tosses it should, in theory, land 500 times heads and 500 times tails, producing the final result:

?125 at the end for every ?1 bet. Obviously, this is pure theory and the situation must be studied more carefully, as must the strategy to adopt.

As we've said, each gambler has ?100, so let's analyze the results of 14 gamblers using the Martingale approach, which increases the stake by a factor x after every loss. Each gambler starts with a different risk percentage and, in particular, for the first it's 1%, the second 2%, the third 3%, the fourth 4%, the fifth 5%, the sixth 10%, then 15%, 20%, 25%, 30%, 35%, 40%, 45%, and 50%.

The first gambler with a factor x = 2 on the first spin risking 1% bets ?1 euro (1% of the ?100 capital is ?1). If he wins, he'll again bet 1% of the new capital ?101.25 (he won ?1.25), which is ?1.0125. If he lost, however, he'll have ?99, and using a factor x = 2, he'll double the initial risk to risk 2%, so

If the gambler wins, he'll go back to staking 1%; he would have won in the previous stake

and would therefore have a capital of ?101.475, of which he'll stake

If he lost, however, he'd have

At this point, he'd stake 4% (double the previous 2%) and bet

and so on.

The second gambler will immediately stake 2% equal to ?2 (2% of ?100) and then proceed using the same logic; the third would start with ?3 (3% of ?100) and the last, daring or reckless, would start by betting ?50 (50% of ?100).

Figure 1.1 shows the results after 100 and 1,000 coin tosses. The simulation produced 53 heads and 47 tails in the first 100 tosses and a total of 467 heads and 533 tails after 1,000 tosses. Note that after 100 tosses, only the gamblers who bet less than 10% still have available funds, while those who started with a greater risk have used up all their capital. The gambler who started with 5% has increased his capital tenfold to ?1,051.98. Note that the gamblers' luck was in; in fact, the wins amounted to 53% of the total. Continuing the game, however, after 1,000 tosses, all the gamblers have lost every last penny, perhaps also due to an unfavourable turn of events that brought the percentage of wins in the first 100 tosses down to 46.7%.