Exness: positive I win, negative you lose

This article will use a simple coin game , Let's learn about the small problems you may encounter in the process of trading , And from a statistical point of view , Give a clear answer .

Suppose your friend asks you to bet while tossing a coin . If the coin is facing up , He will pay you 60% The stakes of . If it's the opposite , You give it to him 40% The reward for . That sounds like a good deal , Because according to his requirements , The average rate of return is 10%.

namely ：(0.6 + 1.6 ) / 2=1.1

If you use 1 The dollar starts , Continuous throwing 100 The second coin , Bet the rest of the money you won or lost in the previous coin toss , Then your average income will be 13780.61 dollar (1.1¹⁰⁰). This sounds like a good proposal .

throw 100 Next time , You should have enough opportunities to play that 10% The advantages of .

Let's do it 1 Ten thousand consecutive throws 100 Second coin bet , Let's see the final result . In the chart below ,y The axis is the result of how much money you have ( For display convenience , We use logarithmic scale ).x The axis represents the number of times the coin is reversed . After simulation verification , It provides a clear view .

Here's the surprise , When you play this coin game ,65% All my time is losing money . The median is 0.22 dollar . You have a quarter chance , You will lose almost all your money ( lower than 0.01 dollar ).

So what happened ?

Why didn't you get your “ Expected return ”? One way to consider the results is , Observe through statistical distribution .

In the seemingly countless paths above , The ultimate benefit of the cap is 2600 Thousands of dollars . This improves the arithmetic mean . If the result of all other paths is 0 dollar , Then this result alone can improve the average value a lot . let me put it another way , This distribution has a long tail (long tail).

Also consider the need for more revenue (%) To make up for the loss . For each 40% The loss of , We need the following 67% The income can return to balance ( The goal is / At present -1= 1/0.6-1). Unfortunately , When the coin falls in the opposite direction , We can only get 60% Revenue .

When the result is compounded by multiplication , The geometric mean is a better average statistic .

The geometric mean is the product of each result , Take the square root of its number . under these circumstances , Your geometric average return is -2%.100 The geometric expected return of this time is 0.13 dollar ($0.98¹⁰)

Geometric mean = (0.6 * 1.6)^(0.5)=0.98

notes ：^0.5 That is, the square root  

Is it worth it on the coin issue ?

Continue to bet your money on this issue , Destined to lose most of the time , So it has become your friend's ATM . In the long run , There are some problems with this bet .

He has 65% Opportunity to earn pocket money , But he has 5% I owe you more than 500 dollar , most 2600 Thousands of dollars . It's certainly best if you don't encounter similar situations .

in summary , It may be worth playing this game with your friends . Although lose 1 There is a great chance of dollars ( It's no big deal ), But there is also a small chance of winning hundreds of dollars ( Or more ). The asymmetry of returns may be worth it .

If your initial bet is forced to be large , This situation becomes unbearable for you . I wonder if we can remedy this by injecting more wisely .

The bet of Kelly formula

The betting concept under Kelly's formula is , You can increase your median by reducing your bet size ( Or geometric average ) Return . It limits your losses , So in the next round , You can still make a reasonable bet . Your bonus will also be reduced , But if you have an average advantage , You may trade a lower average return for a higher median return .

The traditional Kelly bet is a fixed percentage of your potential cash . At odds 、 When both gains and losses are fixed , The best bet size can be calculated .

Bet percentage =( The probability of winning )/( Percentage of loss )-( The probability of losing )/( Percentage of profit )

= 0.5 / (0.4) + (0.5) / 0.6 = 0.4167

We can verify this bet size choice through simulation . I started from 1% Start , It's running again 100 Coin toss 10000 Paths . As expected , The geometric average rate of return is 41.67% The bet size reached a peak near .

Conclusion

With this new knowledge , You can bet with your friends safely . Just remember that , The bet for each coin toss is a little less than half of your money . The following is the optimized capital return compared with the original setting , The same is 10000 Sub simulation .

The bet group optimized by Kelly formula is represented by orange , Obviously, the orange path is tighter , And the distribution center is greater than 1.

In these new circumstances , Your median cash position is 6.16 dollar . And only 0.06% The chance to lose all the money .

Whether in trading or investment , Kelly formula is an important means to optimize fund management , If your transaction is not going well , You might as well redesign your strategy from this perspective .