Kelly Criterion: How can I maximize my Profits in Forex Trading?

Posted on Posted in Education, Money Management

Have you ever heard of Kelly Criterion? In the next 10 minutes, I’m going to discuss this interesting formula to try to maximize your profits in Forex Trading.
Let’s assume that you have the perfect strategy: you win 100% of times. How much are you going to invest? Of course you are going to invest all the money you have, you are sure that you are going to win. Unfortunately such a condition doesn’t exist, you can’t win 100% of times.
So let’s assume now that you don’t always win, but that you win 80% of times. How much are you going to invest? 2%? Less than 2%? More than 2%?

Traders are afraid to invest too much or too little, they always try to find the perfect money management rules that allow them to understand what’s the best percentage of their account to invest. An answer to this eternal question is given by the Kelly Criterion.

Let’s analyze a very easy game before getting into Forex trading. I want to play a simple game just rolling a dice. The rules are:

• If it shows 1, 2, 3 or 4, you win \$10
• If it shows 5 or 6, you lose \$10

Does it look like a good opportunity to you? You have 4 chances out of 6 to win,. Without doing any math calculation, you realize that it is a good opportunity, only by intuition.

From a mathematical point of you, to see if it is a good opportunity and how good it is, you need to calculate the Expected Value (EV):

EV = \$10 x 0.67 – \$10 x 0.33 = \$3.40

0.67 is the probability to win. You have 4 chances out of 6, so 4/6 = 0.67.
0.33 is the probability to lose. You have 2 chances out of 6, so 2/6 = 0.33.

The EV is \$3.40. This means that, in the long run, you should win \$3.40 per every time I roll the dice. So after 100 times, you should be winning \$340.

Now we know, from a mathematical point of view, that this is a good opportunity, the odds are in your favor. How much are you going to bet on it?
Let’s say that you think that it is almost impossible that you can lose more than 4 times in a row, so you want to bet 25% of your account.

I have built an excel file with a function that gives casual numbers from 1 to 6. Let’s see how it goes.

This chart is very interesting.  Let’s analyze it by colours.

RED: the game doesn’t start very well, 3 bad numbers and we are already losing more than 50% of our account.

GREEN: it takes 4 good numbers to go back to more or less the same amount we had at the beginning. This is the first point you should focus on. Why does it take 4 good trades to cover 3 bad trades?
When you lose \$5000 out of \$10000, you lose 50% of your account. To get that money back, you need to make \$5000 with \$5000 (that is your remaining balance), which represents 100%, not 50%.
So you have lost 50%, but to take it back, you need to make 100%… why does it happens? Simply because you have risked too much and, after a couple of trades, you have put yourself in a position in which you need to do extra work to recover your losses.

LIGHT BLUE: 6 winning numbers in a row, the account goes to \$16,093.26, which represents +60.93%. It looks good, but it’s not if you consider that you were losing more or less the same amount after only 3 bad numbers.

PURPLE: after 7 good trades and 5 bad ones, we are almost at the same amount we had at the beginning. There’s something wrong.

YELLOW: the account reaches its peak.

At the end, after 4 bad numbers, we are at -44.16%.

What can we say about this? In an intuitive way, it can look that, if you have a good strategy, the more you invest, the better. Of course, you win more than you lose, so you want to invest a lot because you want to get your big money very soon. This example shows that risking a huge percentage of your account is not always (I would say it’s never) the best way to follow.

Let’s use the same chart, this time investing only 1%:

What was a -44.16% is now -0.0013%. The reason has been already explained, but I’m going to tell you again. When you lose a large percentage of your account, you need a super score to get back what you have lost. This is explained by the so-called percentage recovery formula:

y = x/(1-x)

where y is the percentage gain required to break even and x is the percentage lost.
So if you have lost 20% (0.20), you have:

y = 0.2/(1-0.2)

y = 0.25 = 25%

You need to perform a 25% to recover a loss of 20%.

Loss (%)……. Req’d Gain
5%…………….. 5.2%
10%…………….. 11%
15%…………….. 18%
20%…………….. 25%
25%…………….. 33%
30%…………….. 43%
35%…………….. 54%
40%…………….. 67%
45%…………….. 82%
50%………………….. 100%

This should leave no doubt. The more you lose, the more it will be difficult to get it back.

This first part of the article explains why expert traders are afraid to risk too much.

There’s no need to go into math again to explain why traders are afraid to risk too little. The aim of any investment is to maximize the profit, while keeping the risk under control. If your risk is too low, your potential return will also be very low.

There’s something between “too much” and “too little”, that thing that traders always try to catch, that percentage that would give adequate returns while keeping the risk under control.
The Kelly Criterion is one of the most popular way to try to solve this problem and is one of the most interesting money management strategy.

K = [PxB – (1-P)]/B

Where:

K = Kelly Optimal % Risk
P = Odds of winning
B = Payout

Now let’s calculate the optimal % risk, using the Kelly formula on our previous example.

K = [0.67×1 – (1-0.67)]/1
K = 0.34 = 34%

34%?! It looks like a crazy percentage to risk, isn’t it?
Theoretically, this should be the best percentage to risk, if it wasn’t for something called variance, that will give you some bad streaks during your career as a trader. So from a practical point of view, we need to adjust this calculation.
Here it comes the so called half-Kelly and, also, what I call the ¼ Kelly.

The half-Kelly is very simple to calculate, you only need to divide the Kelly Criterion by 2. In the previous example, we would get a percentage of 17%. Let’s report all of this in a chart:

0% – ½ Kelly = Conservative strategy

½ Kelly – Kelly = Aggressive Strategy

Kelly – 100% = Iper-aggressive strategy

As you can see, from 0% to ½ Kelly, you are using a conservative strategy, that means that your risk is, theoretically, not too high.
From ½ Kelly to Kelly, you are using an aggressive strategy and risking a substantial part of your trading account.
From Kelly to 100%, you are definitely iper-aggressive and are taking a huge risk.

What’s the advantage of the ½ Kelly compared to the standard Kelly Criterion? The answer is very clear on the chart:

On the X-axis we have the risk and you can see that the risk splits in half when we go from the Kelly Criterion to half-Kelly.
The interesting thing that you can notice is that the return decreases, but less than a half.
So the advantage of the half-Kelly is that if you give up a small percentage of your return, you can halve your risk.

I have used the Kelly Criterion and the half-Kelly for my trading. At the end, I decided to use something that I call ¼ Kelly. I simply divide the Kelly optimal percentage by 4.
I did my calculation, I’m not going to write everything here, but it looks that dividing by 4, you can halve again the risk (so you risk ½ of the ½ Kelly), keeping a return that is about ½ of the ½ Kelly.
This is just because I’m very very risk averse and I always follow Warren Buffett’s advice to protect first my money and then try to make profits.

Let’s talk about trading now. A strategy that wins 66% of times is very hard to get in Forex trading, so we can analyze a more realistic strategy that wins 55% of times (that is also a great percentage in my opinion):

K = [0.55×1 – (1-0.55)]/1

K = 0.10 = 10%

½ K = 0.05 = 5%

¼ K = 0.025 = 2.5%

With that kind of strategy, you can decide to be aggressive and risk 10% of your budget per trade; you can be conservative risking 5% per trade or you can be even more conservative risking 2.5% per trade.

What if we win 40% of times but with a risk to reward of 1:2?
In this case P = 0.40 and B = 2, so:

K = [0.40×2 – (1-0.40)]/2

K = 0.10 = 10%

½ K = 0.05 = 5%

¼ K = 0.025 = 2.5%

As you may understand, the Kelly Criterion is entirely based on your trading strategy. In particular on the return that your trading strategy gives over time. So the most important thing is to be extremely accurate when you calculate P. If you have a trading history of 10 trades, the value of P is not going to be reliable, it can easily change after a couple of trades.
Even 100 trades are not enough. A thing is if you say “I won 55 trades out of 100, so my strategy gives me a 55% of winning percentage”; totally another thing is when you say “I won 5500 trades in my last 10000, so my strategy has a winning percentage of 55%.”