**THE FUNDAMENTAL EQUATION OF TRADING**

Copyright Michael Colantoni 2020

R = Risk or loss when a trade fails (average for unsuccessful trades). A negative number.

r = return or profit when successful (average for successful trades).

E = Expectancy over many trades – expected average profit per trade over all trades.

P = total profit over N trades.

N = total number of trades.

F = Failure, the number of losses in N trades.

S = Success, the number of successful trades in N trades.

Success is break even or profit.

Failure is loss.

The fundamental equation of trading is:

**r x S + R x F = P**

N = S + F

E = P/N = Expectancy per trade over many trades.

E = n x R, where n is greater than 0. Preferably, n > 2.

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r is always denoted in terms of multiples of R.

E is always denoted in terms of multiples of R.

The current trade, or the next, is irrelevant. We cannot know the outcome, nor should we care. We are totally objective about the outcome of a single trade. It cannot be predicted with confidence. In fact, we can assume the probability of success is 50%.

R is the core of trading. Everything is measured in terms of R.

R can be expressed in dollars, or as a percentage of Capital, C.

If it is expressed as dollars, where does the dollar value come from? It comes from the percentage of C. C determines how many consecutive failed trades can occur before you are out of business.

R has 2 components – size and probability.The term R represents size, but it has an accompanying probability which is reflected in E. The probability of R is the complement of the probability of r. That is, a 30 per cent probability of losing R, is a 70 per cent probability of gaining r.

When determining how much we will be willing to risk on each trade we have to take into account the probability of failure and use that to determine the size of R ie. the risk we are willing to take on each individual trade.

The probability of failure will vary with different markets. That is, the risk of failure, probability, involved in shares is not the same as that in futures or forex. So our testing must establish distinct R values for each market based on the probability of failure for that market, which is an output from our testing.

Testing our system should yield not just an average failure rate but also some indication of the likelihood of a number of CONSECUTIVE or near consecutive failures such that the drawdown on the account does not require an EXTRAORDINARY success rate to recoup the loss. An extraordinary success rate is unlikely and cannot be relied upon. This is the probability of crashing out. There is some flexibility in the acceptable measure of this set of consecutive failures, but some indication comes from simple arithmetic. If we lose ⅓ of our capital, C, we are left with 2/3C. To get back that 1/3C, we now have 2/3C to work with which means that we have to achieve a profit of 50 percent of C, or ½C. This is just to get back to where we started! A rule of thumb, allowing for personal preferences or trading style, is a drawdown of 10 to 20 percent is the limit. Any system that exceeds that should probably be rejected or modified.

By studying and testing each market we want to trade, we find out such characteristics for our system and adjust R accordingly. This may result in, for instance, the following sets of parameters being adopted.

For futures, we risk R = 2% C

For forex, we risk R = 1% C

For shares, R = 0.5% C

These proportions can also vary depending on the various time frames we want to trade in, as well as time of day or season of the year, and other time related factors. All of these must be tested before trading, and reviewed regularly.

We can avoid the need for this work by only trading one TYPE of market and only one time frame. This is SPECIALISATION. This often goes even further. Many traders are specialists in only a few select instruments in one type of market, even to the point, sometimes, of specialising in a single instrument.

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**We take on Risk so that we may be exposed to the possibility of return, r**.

There is no other way to profit from trading. So we embrace risk, and use it as our measure of expectancy.

E = n x R.

We can often correctly predict the **direction** of the next move, even to 80% or more accuracy. BUT we can rarely reliably predict the **distance** that price will move in that direction. SO we cannot predict the outcome of a single trade, or even of 4 or 5 trades. BUT by testing and by review, we CAN reliably predict the long term outcome of many trades. So we, as traders, are not in the business of forecasting the market, but we ARE in the business of forecasting our EQUITY CURVE.

We should be confident of achieving E over a set of 20, or preferably 100, trades. This predictability is the basis of trading. Anything else is gambling. This is how trading becomes a business rather than a hobby.

HOW do we achieve this PREDICTABILITY?

- Have a set of rules, and NEVER ignore our rules. Consistency is the secret of success.

- Establish a system and test it until we can predict E.

- A system is made up of:
- A set of INITIAL CONDITIONS. Factors that must be true – this identifies the setup. The bar at which ALL these conditions are true is called the SIGNAL BAR – this then is what we call a SETUP.
- FILTERS – factors that are NOT present at the setup.
- ESSENTIAL FILTERS – must NOT be true at the time of the SIGNAL.
- PREFERRED FILTERS – filters that do not disqualify a setup, but we use these to rank potential setups in different instruments to select the one that satisfies the most preferred filters.

- ENTRY TRIGGER – the action that triggers entry into a trade, usually the breaking of the signal bar. It is a determined price in the direction of the intended trade.
- STOP LOSS – the price, usually related to a bar formation, eg. a pivot, that indicates that the initial conditions have been nullified and that the premise of the trade is invalid once this price is crossed in the direction AGAINST the trade direction.
- Risk, R, is the difference between the entry trigger and the stop loss. This is used to measure everything else we do in the trade. We use this to determine the SIZE of the trade position we take so that the total risk in that trade, as a percentage of Capital, C, does not exceed the predetermined limit. That percentage of C has been established through testing and personal preferences. It must be adhered to exactly with no variation. Varying this invalidates E as our Equity Curve is then no longer predictable. Then you are not trading – you are gambling or pursuing a hobby.
- A MANAGEMENT PLAN for staying in the trade to maximise the profit potential offered by the market and for protecting profits as they unfold.
- An example might be – use the stop loss as the exit trigger until we reach a profit of 2R, then move the stop to break even – the entry point. Once we get to 3R,then use the last pivot as the exit point.
- There is never any flexibility in entering a trade – all the factors must be ticked off and they never vary. However, there is flexibility available in exiting a trade EXCEPT in terms of the stop loss. As the trade unfolds, the market gives indications of what it is willing to give you. You do not control what is offered, but you do control your willingness to take it or to persist for a better offer, all the while maintaining your last stop loss as inviolable. Whilst we have the luxury of waiting for a setup to appear before entry, once we are in the trade we must respond to the behaviour of the market. We can have a set ultimate exit point defined by an arbitrary factor such as the break of a moving average, but to ignore the signals being given to you by the market is to ignore the potential for maximum profit.
- INTUITION, or gut feel, can be a valid factor in determining your response to market behaviour. First make sure you have achieved an appropriate level of experience and skill. Never use intuition to enter a trade – never. Never use intuition to exit a trade for a loss – keep to your initial stop loss. But intuition is valid in exiting a trade for a profit, or for moving exit stops closer to price, or for varying your choice of exit triggers. Use of intuition for any other reason is the beginning of crashing out.

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Total Profit is our goal if we are running trading as a business.

Total Profit, P, is a function of E, Expectancy per trade, and N, the total number of trades. And E is a function of R, the risk per trade. And R is a function of Capital, C.

What does this tell us?

It tells us that to increase P we must increase R.

There are only two ways to increase R.

Either take on a greater percentage risk per trade, or increase C.

Increasing the percentage risk also increases the risk of crashing out and is therefore unacceptable. So the only way to increase P is to increase C.

This can only be done one of 2 ways. Either add more capital to your account out of other savings that you are willing or able to risk. Or, regularly review your account size and change your value for C to include profits that have been made over the last period.

The latter is the method used by professional traders who trade their own accounts. It also means that, if you have made a loss in the last period, C decreases and therefore the likely P gets smaller.

But on the assumption that you are trading successfully, C should increase each period and thus your potential for profit increases with it. So we compound our profits at the end of each period to increase the dollar risk taken on each trade reflecting the new size of our account, C.

The periods used are either monthly or quarterly. Most common among individual traders is probably a monthly period. We take the realised profit at the end of each period and add it to the C we were using in the prior period. From this we deduct the potential loss in any trades that are still open as we do not know the outcome of those trades. This gives us a new C on which to base the R we will use for sizing future trades in the next period.

To see the effect of this over time, we denote the increase in C, the compounded profit, in terms of a percentage of C for each period.

We find that, if we can average a 6% profit in each month, by the end of the year we will have doubled our account size.

If we can average a 10% profit in each month, we find that we will have tripled our account size by the end of the year.

Let’s say we start with a $20,000 account. C = 20,000.

A 6% profit is $1200. That does not seem too hard to achieve and most reasonably competent traders seem able to achieve that. But that means that by the end of the year, you will have an account size of $40,000.

If we leave out tax considerations for a moment, this means that the following results are achievable.

Year 1 $40,000

Year 2 $80,000

Year 3 $160,000

Year 4 $320,000

Year 5 $640,000

Year 6 $1,280,000

Starting with $20,000, you become a millionaire somewhere between year 5 and 6. If you start with $50,000, this will be achieved soon after year 4.

With some experience, a 10% monthly return seems possible given the experience of many traders. This triples your account each year giving the following results from a $20,000 start

Year 1 $60,000

Year 2 $180,000

Year 3 $540,000

Year 4 $1,620,000

Year 5 $4,860,000

Obviously, this will require trading in very liquid markets to be able to place the size of trade required to maintain the returns needed.

BUT we do have to pay tax, and this significantly impacts on the numbers above. Each year we have to deduct the tax payable – but during that fiscal year, we can compound the full amounts. However, the results, assuming a 33% average tax rate (for example), results in:

Starting with $20,000 and achieving 6% per month:

Year 1 $33,000

Year 2 $49,000

Year 3 $81,000

Year 4 $134,000

Year 5 $223,000

Year 6 $375,000

Year 7 $625,000

Year 8 $1,042,000.

Don’t you just hate that tax man?

The effect of the tax man has added a bit over 2 years to the time required to achieve your first million dollars. But that is starting with a paltry $20,000 account. Starting with a more reasonable exposure of a $50,000 account results in a $2,600,000 account in the same 8 years or millionaire status in a little over 6 years.

Many successful traders achieve returns each month of well over these figures. 20 or 25% per month happens. Work out the returns from that sort of performance!