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Money management is one of the fundamental pillars (in addition to psychotrading, system and risk management) for trading to work. It is the key to consistency and profitability over time as its function is to protect your account by maximising returns and minimising risk by defining how much to buy and how much to sell depending on the capital you have available. Therefore, it is operational risk management (which is different from risk management, which is in charge of setting the stop loss), which calculates the size of the next position.

With good money management we will achieve that our account loses as little as possible when we have losing trades and that profits increase over losses when we have winning trades. In other words, earning more with less risk, which is known as the risk/return ratio.

## Risk / reward ratio

The risk/reward ratio measures the risk you have to take in order to receive a profit. A risk/reward ratio of 1:2 means that when you get it right, you earn twice as much as you lose. A risk/reward ratio of 3:1, on the other hand, means that when you get it right you win one, but when you miss you lose three times as much as you win. This could lead to ruin. Finally, a 1:1 ratio means that you win as much as you lose, so if you have a 50% success rate (you win 50% of the time) you can’t use a 1:1 ratio, it must be higher. The ratio always goes hand in hand with the success rate of a trading system.

The formula for the risk/return ratio is:

**Ratio= total losses/total profits.**

Before opening any trade, you decide what your stop is. The distance between your entry and stop loss is your risk, and the distance between your entry and stop profit is your reward.

Knowing this ratio helps us to:

1. Decide which trades we will make, based on the past results we have achieved.

2. Choose only those trades that offer an appropriate reward to risk ratio.

The expected return must be in accordance with the risk to be assumed and also with the mathematical expectation of our trading system.

## The mathematical expectation

It is a statistical concept that, when applied to trading, tells us whether a system is a winner or a loser because it shows us the average amount of profit or loss that can be expected over the long term. It is one of the best statistics because it quantifies the performance of a strategy or system, which is independent of the size of the capital.

The formula is:

**MS=(Percentage of winning trades X average profit)-(Percentage of losing trades X average loss).**

This formula can give us the following results and are interpreted as follows:

- Expectation > 0. It is positive, indicating that he earns more than he loses on average.
- Expectation = 0. It is neutral, indicating that your gains are equal to your losses on average.
- Hope < 0. It is negative, indicating that he loses more than he gains on average.

From the mathematical expectation we draw the following conclusions:

1. To make money you don’t have to win every trade, the important thing is the combination between how many times you get right (or wrong) and how much you win per trade.

2. To be profitable the mathematical expectation must be positive, and the higher the better.

3. Positive mathematical expectation does not prevent losing streaks, nor does it say anything about volatility in results.

Let’s look at an example. Suppose we have the following data after a few demo trades:

- Percentage of winning trades: 75%
- Average profit: 100
- Percentage of losing trades: 25%.
- Average loss: 200

We do the mathematical expectation which would be: **(100*0.75) – (200*0.25) = 25.**

As it gave a positive result, we can see that the system or strategy of the example is a winning system.

Let us now look at the opposite example. The data collected are:

- Percentage of winning trades: 60%
- Average profit: 100
- Percentage of losing trades: 40%.
- Average loss: 200

We do the mathematical expectation which would be:** (100*0.60) – (200*0.40) = -20.**

The mathematical expectation is negative, so in this case it is a losing system.

Note that it does not matter if the probability of success, i.e. the rate of winning trades, which in this case is 60%, is higher than the rate of losses; with a negative expectation it is still a system that only loses money in the long run.

Now let’s look at an example with respect to the probability of success. Suppose we collect data from several demo trades and get the following metrics:

- Hit probability (Percentage of winning trades): 25%.
- Average profit: 400
- Probability of failure (Percentage of losing trades): 75%.
- Average loss: 100

The mathematical expectation would be: **(0.25*400)-(0.75*100)= 25**

The average per trade of this system earns exactly the same as the first example. But, although they have the same mathematical expectation, comparing the probabilities of hit and miss we can see that the first system hits more than the second, but has a lower average win and a higher average loss.

## Which one to choose or which one is better?

The system in the first example has a higher probability of success, therefore it will cause a lower volatility in our capital, which reduces the risk. For this reason it is always recommended to choose a system or strategy that has a higher probability of success, even if it has a lower average profit, even if compared to another strategy it has the same mathematical expectation.

As a conclusion we can say that the rate of winning trades and the risk/reward ratio alone are not so important. What is important is what happens when the two are combined to determine the mathematical expectation of a strategy. If a trader has a mathematical expectation of +3, it means that on average, he can make 3 times the amount he risks on each trade he makes using his strategy or system.

**Other factors to consider**

In addition to knowing the mathematical expectation of our system, there are other important factors that come into play. Below we will look at some concepts to consider when it comes to evaluating a trading system:

## Historical results

Whenever we evaluate the hopefulness of a trading system it is always done with historical data, i.e. trades that have already been made. While this is important for demo testing, detecting errors and improving the strategy, we must not forget that there is no guarantee that we will have the same positive results in the future. We must be careful to avoid over-fitting our approach to historical data.

## Commissions:

Commissions and similar charges may seem small on a single trade, but when large numbers of trades are made with a system that has a small positive expectation, these costs can eat into profits or even turn the expectation from positive to negative. For this reason, the positive expectation must be large enough for commissions to be included.

## Position size:

Position size and mathematical expectation go hand in hand because, even with a large positive expectation, the use of an erratic position size can change the results and jeopardise your trading account. Position sizing should be kept within tolerable levels. We will expand on this point in the next lesson.