One of the most important formulas in investment-Kelly formula

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Drunk dancing by pavilion, half a book, sitting and talking about the sky.

Those things in the stock market column

What is Kelly formula?

Kelly formula is a gambling strategy that makes the long-term growth rate of principal the fastest in independent and repeated gambling games. "Independent repetition" emphasizes the independence and irrelevance between each bet, while "long-term" emphasizes the exclusion of accidental factors and looks at it from the perspective of probability.

What is the content of Kelly formula?

Suppose you make an investment, the probability of winning is p, the probability of losing is (1-P), the percentage of profit when winning is a, and the percentage of loss when losing is b, then the optimal investment position c is:

So is this formula correct?

We use forgotten mathematical knowledge to make a proof:

If you invest n times, and the percentage of each position is H, you gain M times and lose N-M times, and the initial capital is Z0, then the total assets Zn after n times of investment is:

Does this formula make you dizzy? Then let's beautify it (both sides take logarithm at the same time) and it becomes:

According to senior high school math problem-solving experience, further deformation, anyway, is to play:

Then, according to the knowledge of seeking the limit of higher mathematics in universities, when n tends to infinity, M/n = P, (N-M)/n = (1-P), then the above formula becomes:

Then, using the knowledge of the derivative of higher order numbers in universities, the second derivative of the above function to H is less than 0 (the reader proves it by himself, but only consolidates his own mathematical knowledge), so when its first derivative is equal to 0, the maximum value can be obtained. Its first derivative is:

Then, make it equal to 0, both:

Finally, after simplifying with junior high school knowledge:

The card must be!

Obviously, the formula P*a-( 1-P)*b is the expected e.

Let b be 1, which means the loss of principal, so there are:

To put it simply: as long as there is a probability of losing money (1-P), no matter how big a (profit) is, don't invest more than p, in other words, as long as the probability of losing money is not 0, you must not Man Cang; As long as the probability of losing money reaches 0.5, it must not exceed half a warehouse-even if there is a possibility of 10 and 20 times profit. This conclusion is particularly meaningful for investments such as futures, options, warrants and loans. Many people lose a lot in futures, options, foreign exchange and other transactions, or lose all their money after playing for a long time. The reason is that they are tempted by possible high profits and their positions are often too large [1].