This is a transcript of the video that you can access here.

Hello everyone, welcome back to the NM quantitative trading tutorials. Today I am going to show you why you always lose money in the stock market. When you lose money, you think either you are not smart enough or just bad luck. The reality is that it may have nothing to do with whatever you do. The sad truth is that you would and will lose money anyway.

Before that, I am going to teach you a fun game that you can play to pick up girls in bars. Fun huh? Let’s say you see a girl that you like and need an icebreaker. Suppose you look like Thor M-sworth, you can try to offer her to play this game. You both get a coin. Each round you both show either a head or a tail. If you both show heads, you pay her $3. If you both show tails, you pay her $1. Otherwise, if you two show different sides of the coins, she loses and pays you $2. Since you look like Thor, she probably will agree to play. It seems like a fair game anyway. A simple mind would think that 3 plus 1 minus 2 minus 2 equals 0. So, we have the following payoff matrix.

M/P | + | – |

+ | 3 | -2 |

– | -2 | 1 |

You are M for M-sworth and She is P for Player.

As she keeps playing, she will realize that she keeps losing money. Why? Let’s analyze this game using mathematics. First, she can’t keep choosing heads or tails. If she always chooses heads, for example, then you can always choose tails so she loses. She must be sometimes playing heads and sometimes playing tails. She must play a probabilistic strategy, sometimes heads and sometimes tails to keep you guessing. Likewise, you must also be playing a probabilistic strategy to win as much as you can to impress her. The question is then how often you play heads and how often you play tails to maximize your profit or, equivalently, to minimize her profit.

Let \(y\) be the probability that you play heads and \(x\) the probability that she plays heads. In terms of our playoff table, the probability of M equals plus is \(y\).

\(P(M = +) = y\)The probability of P equals plus is \(x\).

\(P(P = +) = x\)Now we can compute the expected profit of the player. When you both play heads, the probability is x y, the payoff is 3 dollars. This is the first term in the equation. When you both play tails, the probability is 1 minus x, 1 minus y, the payoff is 1 dollar. The probability of you playing tails and she playing heads is x times 1 minus y, the payoff is 2 dollars. Similarly, the probability of you playing heads and she playing tails is y times 1 minus x, the payoff is also 2 dollars.

\(E(P) = 3xy + 1(1-x)(1-y) – 2x(1-y) – 2(1-x)y = 3xy + 1 – x – y + xy – 2x + 2xy – 2y + 2xy = 8xy -3x – 3y +1\)Expanding the terms and regrouping, we have 8 x y minus 3 x minus 3 y plus 1.

Since you want her to lose money so then she has to do whatever you ask, you want the expectation of P to be negative.

\(E(P) = 8xy – 3x – 3y + 1 < 0\)We have this inequality. 8 x minus 3 times y is less than 3 x minus 1

\((8x – 3)y < 3x – 1\)We want to write y, your probability of playing heads that you can control, in terms of x.

To manipulate inequality, we need to consider two cases.

The first case is when 8 x minus 3 is positive. This is equivalent to x is bigger than 0.375. We move the term to the right. Then we have:

\(y < \frac{3x – 1}{8x – 3}\)We can plot this graph using S2.

The source code is here:

The executable is here:

The plot is:

The y probability goes from 1 to the lowest of 0.4 when x goes to 1. When x equals 1, y equals 0.4. It means that as long as you play your probability strategy below the curve, the red region, you will always make money in the long run. Your opponent, the player, she will always lose money in the long run, no matter what she does! In fact, you don’t even need to know what x is, you don’t even need to know her strategy, you just need to play y smaller than 0.4. Then you always win!

The second case is when 8 x minus 3 is negative. This is equivalent to x is smaller than 0.375.

When we move the term to the right, we need to flip the inequality sign. Then we have:

\(y > \frac{3x – 1}{8x – 3}\)We can again plot the graph.

The y probability goes from one-third, the highest, to 0, depending on what her probability of playing heads is, that is x. When x equals 0, y equals one-third. Similarly, as long as you play above the curve, you will always win in the long run. Again, you don’t even need to know what x or what her strategy is as long as you play y above one-third.

Now when we combine the two cases, there is a region that you always win regardless of what x is. That is when y is bigger than one-third and smaller than 0.4. When y is in this interval, you, the M, always make money in the long run regardless of what x is or what strategy the player plays. Her expected profit is always negative when y is in this interval! It is impossible for her to win.

We can apply the same concept to the stock market. While it seems that the stock market is a game of intelligence or luck because the market goes up and down randomly all the time. We just have to “guess” the trend correctly to make money. Not quite so if we put the stock market guessing game in the perspective of market maker versus player. The player only makes money if she follows the trend of the market maker of the stock. That is, if the market maker or big players wants to drive the price up and the player longs the stock, then she makes money. If the market maker wants to drive the price down and the player shorts the stock, then she also makes money. In other words, she or you only make money when you follow the market makers. On the other hand, if you flip a different side than the market makers or big players, then you lose money. I just show you above using mathematics that it is impossible for you to make money if the market makers play their (probability) strategy right and they will! The market makers, like the casinos, are not here to provide you with free money or to make your life better. Quite the contrary! The reality is that market makers can always pump and dump the stock in a way that you will keep losing money till you have nothing left! It was and is and will never be a fair game for retail investors like yourself!

The (3.0, -2.0, -2.0, 1.0) payoff is not the only payoff matrix that M can have a guaranteed win. Other payoffs are also possible. For instance, (6.0, -5.0, -5.0, 4.0) is another possibility.

```
val plot2 = plot_y(6.0, -5.0, -5.0, 4.0)
plot2
```

The output is:

`y interval = (0.4444444444444444, 0.45454545454545453)`

The plot is:

Any y between 0.44 and 0.45 will guarantee an infinite profit in the long run.

You can try different inputs to the plot_y function to come up with a different payoff matrix and a different (probabilistic) winning strategy. Now, you go play this rigged game in a bar to pick up girls, or try to play it with your friends to make money off them (and to lose friendship)!

Last but not least, I have two pieces of advice for you. First, spend your money and time on something else meaningful like your family and friends. Don’t gamble! Second, if you really must invest your money in the stock market, like, share and subscribe. I will make more videos on how to do quantitative trading and quantitative finance. Hope you find mathematics useful!

Happy trading!