The problem that Kelly's formula aims to solve

Suppose gambling game 1: the probability that you win is 6%, and the probability that you lose is 4%. The net rate of return when winning is 1%, and the loss rate when losing is also 1%. That is, if you win, you can win 1 yuan for every bet on 1 yuan, and if you lose, you will lose 1 yuan for every bet on 1 yuan. Gambling can be done unlimited times, and the bet you make each time is up to you. Question: Assuming that your initial capital is 1 yuan, how do you bet, that is, what percentage of the principal is each bet, to maximize the long-term benefits?

For this gambling game, the expected return of each bet is 6%*1-4%*1=2% of the bet amount, and the expected return is positive. In other words, this is a gambling game that is dominant for gamblers, and it has a very large advantage.

so how should we make a bet?

if we don't think carefully and imagine roughly, we will feel that since the expected return of each bet is 2%, I should try to put more principal in each bet in order to achieve the maximum long-term return. The maximum value of this ratio is 1%.

But obviously, it is unreasonable to put 1% of the principal into every gambling game, because once a gambling game is lost, all the principal will be lost, and you can no longer participate in the next game, so you can only leave in a daze. In the long run, losing a bet is bound to happen, so it is bound to go bankrupt in the long run.

So here comes a conclusion: As long as there is the possibility of losing all the principal at once in a gambling game, even if this possibility is very small, then you can never Man Cang. Because in the long run, small probability events are bound to happen, and in real life, the actual probability of small probability events is far greater than its theoretical probability. This is the fat tail effect in finance.

go back to gamble 1.

since it is unreasonable to bet 1% every time, how about 99%? If you bet 99% every time, you can not only guarantee that you will never go bankrupt, but you may also realize great benefits if you are lucky.

is this the actual situation?

let's not analyze this problem theoretically, but we can do an experiment. Let's simulate this gambling game and bet 99% each time to see what happens.

This simulation experiment is very simple, and it can be completed by excel. Please see the following figure:

As shown above, the first column indicates the number of bureaus. The second column is the winner, and excel will generate 1 with a probability of 6%, that is, 6% of the probability net income is 1, and 4% of the probability net income is -1. The third column is all the funds of gamblers at the end of each game. In this experiment, the position of each bet is 99%, and the initial principal is 1, which are marked in yellow and green respectively.

As you can see from the figure, after playing 1 games, the number of games won in 1 games is 8, which is greater than the probability of 6%, and only lost twice. But even so, the final fund is only 2.46 yuan, which is basically lost.

When I increased the number of experiments to 1,, 2,, 3, ..., the result was predictable. In the end, the funds in my hand basically tended to zero.

Since 99% doesn't work, let's try several other ratios, as shown in the following figure:

As can be seen from the figure, when the position is gradually reduced from 99% to 9%, 8%, 7% and 6%, the result of the same 1 games is completely different. It seems to be seen from the figure that as the position gradually becomes smaller, the funds after 1 games gradually become larger.

When you see this, you will gradually find that the gambling problem is not that simple. Even if the gambler dominates such a big gamble, it is not easy to win money.

so how can you bet to maximize the long-term benefits?

as shown in the above figure, the smaller the ratio, the better? I don't think so, because obviously you can't make money when the ratio becomes zero.

so what is the optimal ratio?

this is the problem that the famous Kelly formula has to solve!

introduction of Kelly formula

where f is the optimal bet ratio. P is the probability of winning. Rw is the net rate of return when winning, for example, in gamble 1, rw=1. Rl is the net loss rate when losing, for example, in gamble 1, rl=1. Note that here rl> 。

According to Kelly's formula, it can be calculated that the most favorable bet ratio in Game 1 is 2%.

We can do some experiments to deepen our understanding of this conclusion.

as shown in the figure, we set the positions as 1%, 15%, 2%, 3% and 4% respectively. Their corresponding columns are D, E, F, G and H respectively.

When I changed the number of experiments to 3, as shown below:

When I changed the number of experiments to 5, as shown below:

As you can see from the two pictures, the result corresponding to column F is the largest, which is not an order of magnitude compared with other columns. The position ratio corresponding to column F is exactly 2%.

You see the power of Kelly's formula. In the above experiment, if you unfortunately choose 4%, that is, corresponding to column H, then after 5 games of gambling, although your principal has changed from 1 to 22799985.75, the income is huge. But compared with the result of 2%, that's really equivalent to not making money.

this is the power of knowledge!

Understanding Kelly Formula

The mathematical derivation of Kelly Formula is very complicated and requires very advanced mathematical knowledge, so it is meaningless to discuss it here. Well, to put it bluntly, even I don't understand it. Here I will deepen your subjective understanding of Kelly's formula through some experiments.

let's watch another gambling game. Bet 2: Your chances of losing and winning are 5% respectively, such as flipping a coin. When winning, the net return rate is 1, that is, rw=1, and when losing, the net loss rate is .5, that is, rl=.5. That is to say, for every dollar you bet, you can win 1 yuan again when you win, and you only need to pay 5 cents when you lose.

It is easy to see that the expected return of Gambling 2 is .25, which is another gambling game where gamblers have great advantages.

According to Kelly's formula, we can get the best bet ratio for each game:

That is to say, half of the money is bet at a time, and we can get the biggest profit in the long run.

next, I will get the concept of average growth rate r according to the experiment. First of all, let's look at experiment 2.1, as shown in the following two figures:

These two figures are both experiments made by simulating gambling game 2. In the winning and losing column in the second column, the experiment will generate 1 with a 5% probability, which means that the profit is 1%. There is a 5% probability of -.5, which means a loss of 5%. The third and fourth columns are the funds owned after each gambling under the position of 1% and 5% respectively.

a careful comparison of the two figures reveals the first conclusion, that is, after the same number of games, the final result is only related to the number of games won and the number of games lost in these games, and has nothing to do with the order of the games won and lost in these games. For example, in the last two pictures, four games were played, and two games were won and two games were lost in each picture, but the winning and losing order in the first picture was winning and losing, and the winning and losing order in the second picture was winning and losing. They all end up with the same result.

Of course, this conclusion is very easy to prove (multiplication and commutation law, pupils can do it), so I won't prove it here. The above two examples are enough for everyone to understand well.

So since the final outcome has nothing to do with the order of winning or losing, let's assume that Gambling 2 will continue as in Experiment 2.2, as shown in the following figure:

We assume that the outcome of gambling is alternating, which has no effect on the outcome funds in the long run.

let's make a definition before observing the picture by ourselves. Assuming that a certain number of gambling games are regarded as a whole, the frequency of various results in this whole is just equal to its probability, and the number of games in this whole is the smallest among all the qualified whole, then we call this whole a group of gambling games. For example, in the experiment shown above, a group of gambling games represents two gambling games, in which one wins and one loses.

Look carefully at the numbers marked in blue in the picture above, which are the end of a group of bets. You will find that these figures have maintained a steady growth. When the position is 1%, the growth rate of the blue marked number is %, that is, the growth rate of the principal after a group of gambling is %. This also explains that when Man Cang bets every time, it is impossible to make money in the long run in Gambling 2. When the position is 5% (i.e., the optimal ratio obtained by Kelly formula), the growth rate of the blue marked figure is 12.5%, that is, the growth rate of the principal after a group of gambling is 12.5%.

this is a general rule, and the growth rate after each group of gambling is related to the position. And the greater the growth rate after each group of gambling, the more the final income will be in the long run.

according to the growth rate of each gambling group, the average growth rate g of each gambling group can be calculated. In the above figure, each gambling group contains two gambling games, so the average growth rate of each gambling game < P > In fact, this R can be calculated by formula.

in the long run, if you want to maximize capital growth, you just need to maximize R, that is, maximize G.. The best bet ratio F is actually obtained by solving max(g).

other conclusions of Kelly formula-about risk

Kelly legend (this section is from the internet)

Kelly formula was originally at & John Larry Kelly, a physicist at T Bell Laboratories, was founded according to his colleague Claude Elwood Shannon's research on long-distance telephone line noise. Kelly solved the problem of how Shannon's information theory should be applied to a gambler with inside information when betting on horses. The gambler wants to decide the best bet amount, and his inside information doesn't need to be perfect (no noise) to give him a useful advantage. Kelly's formula was then applied to blackjack and the stock market by another colleague of Shannon, edward thorp.

Thorpe used his spare time to write a mathematical paper entitled "blackjack optimization strategy" through months of hard calculation. He used his knowledge to "surprise" all casinos in Reno, Nevada overnight, and successfully won tens of thousands of dollars from the blackjack table. He is also the originator of quantitative trading hedge funds on Wall Street in the United States, and initiated the first quantitative trading hedge fund in the 197s. In 1962, he published his monograph "Beating Bankers", which became one of the classic works of finance.

using prospect

how to make money in real life by using Kelly formula? That is to create a "gambling game" that meets the application conditions of Kelly formula. In my opinion, this "gamble" must come from the financial market.

I have been doing research on trading system recently. What is the most important thing for an excellent trading system? A trading rule with positive expected returns accounts for 1% of the importance, while a good fund control method accounts for 4% of the importance, and the remaining 5% is the psychological control of the manipulator.

and Kelly formula is a sharp weapon to help me control my capital position.

For example, I developed a stock trading system before, which trades once a week, and the probability of success and failure is .8 and .2 respectively. When you succeed, you can earn 3% (deduct commission and stamp duty) and lose 5% every time you fail. Before I knew Kelly's formula, I was blindly trading in Man Cang, and I didn't know if my position was set correctly, so my psychology was very empty. After applying Kelly formula, the best position should be 9.33, that is to say, if the loan interest rate is , you need to use leveraged trading to get the fastest capital growth rate. Through formula calculation, the average growth rate of each transaction is about 7.44%, while the average capital growth rate of Man Cang trading is about 1.35 (actually, it is expected income). Through the experimental simulation, it is also found that the growth rate of leveraged trading funds is much faster than that of Man Cang. This also gives me a better understanding of why many quantitative investment fund companies need to use leveraged trading.

Of course, Kelly formula can't be so simple in practical application, and there are still many difficulties to be overcome. For example, the cost of capital required for leveraged transactions, for example, in reality, capital is not infinitely separable. For example, in the financial market, it is not as simple as the simple gambling mentioned above.

But in any case, Kelly's formula shows us the way forward.