#### The Money Management Task

We have arrived at the key task of trading job: Managing wealth through position size. But to effectively do this critical task, you need to control everything around your trading system. Why? Because this task must be based on the statistical properties of your strategy. We are going to show you why.

#### The trading size is Key

Having a winning system is not enough to succeed. It is the beginning of it, but we can fail without a proper concept of the influence of position size in the overall growth of the trading account, and also on the expected drawdowns.

#### Optimal f

Ralph Vince in his book The Mathematics of Money Management shows the power of proper position size with a coin-toss example, where the gambler is awarded twice his bet on heads while only loses the bet on tails. This game is a clear winner since the gambler has an expectancy of 0.5 cent profit per dollar bet.

His concept of optimal f came from a paper by John Larry Kelly Jr., a Scientist working by the Bell Labs A new interpretation of the Information Rate, published back in 1956. For a more in-depth explanation of the Kelly Criterion, please read its Wikipedia article.

The graph above is the plot of the first 100 occurrences of this coin toss game using different bet sizes. The chart is in log10 scale and the starting balance is $100 (10^2). The figure shows all trading sizes from 1% to 100% in steps of 1%. On the figure, ruin is defined as losing 75% of the initial capital.

**From this figure, we can extract many interesting conclusions:**

- The drawdowns, as seen as sawteeth in the curves, grow as the position size grows.
- There is a per cent amount beyond which there is no more capital growth, but drawdowns continue growing.
- Higher bets beyond the optimal fraction carry the risk of ruin with absolute certainty, even when the game is a net winner. (see the horizontal lines at the bottom of the graph).

Of, course different game types have a particular optimal position size. On the game described above the optimal position, size is betting 25% of the currently available funds. The downside of the optimal bet is it can show huge drawdowns beyond 80%. Therefore we should think about position size differently, especially if you’re starting the trading profession.

#### Position Size as the means to achieve Our Objectives

Dr. Van K. Tharp in his books, see position sizing as the way to achieve financial objectives. Let’s see why.

#### Risk and R

In his well-known book **Trade your Way to your Financial Freedom**, Van K. Tharp states that the key feature of trading success is that the trader should always identify his initial risk before entering a position.

He recommends that this risk should be normalised, and he calls it R. Profits must, also, be computed as a multiple of R, the initial risk.

The risk on an asset is a direct calculation of the difference in points, ticks, pips from the entry point to the stop-loss level multiplied by the cost of that pip or lot.

Consider, for example, the risk of a micro-lot of the EUR/USD pair in the following short entry:

**Size of a micro-lot:** 1,000 units

**Entry point:** 1.19344

**Stop-loss:** 1.19621

**Distance from entry to stop loss**: 0.00277

**Dollar Risk for one micro-lot:** 0.00277 * 1,000 = $ 2.77

In this case, if the trader had set the dollar-risk on every trade to $100, what should be his position size?

–** Position size:** $100/$2.77= -36 micro-lots

Using this concept, you can normalise your position size and adapt it to the individual risk of the trade, which depends, as we have seen, on the distance between the entry point and stop-loss level, a distance that is changing from trade to trade. For instance, if the next trade’s risk were $5, the next position size would be: $100/5 = 20 micro-lots.

That way, the trader enters his position using a standard and controlled risk, independent of the distance from the entry point to stop-loss level.

#### Profit targets as Multiples of R

Profits can be normalised to multiples of the original risk R. Then, it does not matter if you change your dollar risk from $100 to $150. If you keep your records using R multiples, you’ll have a standard track record of your system.

Starting with thirty results, you can discover how a system performs and, also, measure its statistical properties.

Values such as Expectancy (E), mean reward to risk ratio(RR), % of gainers, the number of R-gains a system delivers (R multiple) in a day, week, month or year.

Knowing these figures is very important because it will be used to define the trader’s objectives.

#### The Profitability Formula

The key feature that defines if a system is good or not is expectancy E (the expected value of trades). Expectancy is the expected value of winners (E+) less the expected value of losers (E-)

**(E+) = Sum(G)/(n+) * %Winners**

**(E-) = Sum(L)/(n-) * %Losers**

where

**Sum(G):** The total dollar gains on our sample history, excluding losers

**Sum(L):** The total dollar losses on our sample history, excluding winners

**(n+):** The number of positive trades(Gainers)

**(n-):** The number of negative trades(Losers)

The expectancy** E** then is:

**E = (E+) – (E-)**

When you have your trading record normalised as a list of R-multiples E means the expected R-fraction profit per trade. The beauty of this is that, together with the average number of trades, expectancy normalised to R-multiples can tell the return the system delivers in a time interval.

For example, let’s say your system takes 6 trades per day and its R-adjusted E is **0.45R**.

This means it makes $0.45 per dollar risked on each trade. That means, also, that the system returns an average of 0.45Rx6=2.7R per day, and 54R monthly (20 trading days).

#### Setting Objectives

Let’s say you are using this system and your objective is to reach a monthly income of $5,000. What would you need to risk on every trade?

To answer this, we need to equate 54R = $5000

Therefore your trade risk should be:

** R= 6000/54 = $92.6**

Now you know, for instance, that to achieve $10,000/month you need to double your trade risk to $185.2, and you could get $20,000 if you could raise your trade risk to $370.4. You are able to convert your system into a scalable money making machine, but with a risk-controlled attitude.

#### Variability of the Results

As a conscious trader, you would like to also know what to expect from the system in terms of drawdowns, which, as we know is directly related to position size. Here, in this section, we will understand why.

Is it usual for my system to show 6, 10, 15 or 20 consecutive losses? What is the likelihood of a string of losses? Is my system misbehaving or is on track?

That can be addressed, using the % of losers (**Pl**). Statistical theory tells that the probability of an event A and an event B happening together is the probability of A times the probability of B:

**ProbAB = ProbA * ProbB**

To compute the probability of an n-streak of losses, we simply multiply the probability of a loss by itself n times.

**Prob_Streak_n = Pl ^{n}**

As an example, the probability of two consecutive losses for a 50% losing system is:

**Prob_Streak_2 = 0.5 ^{2 }= 0.25 or 25%**

And the probability of suffering 4 consecutive losses will be:

**Prob_Streak_4 = 0.5 ^{4}= 0.0625 or 6.25%**

And six consequtive losses :

**Prob_Streak_6 = 0.5 ^{6}= 0.015625, or 1.5625%**

And so on.

This result is in direct relation to the probability of ruin. If your risk R is such that a string of six losses wipes 100% of your capital, *there is a probability of 1.56% of that to happen under this system.*

Now you have learned that you must set your dollar risk R to an amount such that a string of losses doesn’t bring the account beyond the maximum per cent drawdown that is tolerable to the trader.

What happens if the system has 40% winners and 60% losers, as is usual on high risk to reward systems? Let’s see:

Prob_Streak_2 = 0.62 = 36%

Prob_Streak_4 = 0.64 = 12.96%

Prob_Streak:6 = 0.66 = 4,66%

Prob_Streak_8 = 0.68 = 1.68%

Prob_Streak_10 = 0.610 = 0.60%

We observe that the probability of consecutive streaks of the same magnitude increases. This means that with systems with a lower per cent of winners, you should be more careful and reduce your maximum risk compared to a system with higher winning ratios.

As a rule of thumb, you must consider that sometime in your trading future you may experience a string of 10 or more losses, and consequently use a trading size which, when that happens your account’s drawdown won’t go higher than tour allowed maximum drawdown.

As an example, let’s do an exercise to estimate the maximum dollar risk for this system on a $10,000 account and a maximum tolerable drawdown of 30%, assuming the trader wanted to withstand 10 consecutive losses (an event with 0,60% probability).

According to this, we will assume a streak of ten consecutive losses, or 10R.

30% of $10,000 is $3,000

then 10R = $3,000, and

max R allowed is: 3000/10 = $300 or 3% of the account balance.

As a final warning, to get a precise measurement of the percentage of losers, you should have to register in excess of 100 live trade samples of your trading history. Just 30 data points is not representative enough to get any precise result.

#### Key Learning Points And Conclusions

- Trading higher than the optimal fraction your strategy allows will only increase your Maximum drawdown and risk of ruin, but not your trading returns.
- Position Size is the part of the system that will determine how much to risk on a trade
- Position Size is, also, the right tool to help you set your financial goals.
- To decide about your optimal risk R you should first measure, register and be aware of the main statistical parameters of your system: Expectancy, per cent gainers and losers, Reward to risk ratio, the mean R etc.
- Position Size has a direct relationship to the size of the Maximum Drawdown.
- You should compute the maximum R allowed by your system, account size and the max drawdown bearable to you and not trade higher than that amount.