Kelly's rule is to bet a predetermined asset score, which may be counterintuitive. In one study, each participant received $25 and was asked to bet on a coin to spend 60% of his time on land. The highest prize is $250. It is worth noting that 28% of the participants went bankrupt, and the average payment was only $965,438 +0. Only 2 1% of the participants reached the maximum. 18 gambled everything on 6 1 participants, and two-thirds of the gambling tails were doing experiments. Neither method is optimal. "Using Kelly standard, according to the odds in the experiment, the correct method is to bet 20% for each shot (see the first example below). If you lose, the size of the bet will be reduced; If you win, the stakes will increase.
Although the promise of Kelly's strategy seems to be more convincing than any other long-term strategy, some economists advocate strong opposition, mainly because individual specific investment constraints may mask the desire for optimal growth rate. Conventional alternative expected utility theory says that the size of the bet should maximize the expected utility of the result (a person's logarithmic utility, Kelly's bet has the greatest expected utility, so there is no conflict; In addition, Kelly's original document clearly stipulates that the gambling game needs practical functions within a limited time. Even Kelly's supporters usually advocate the score Kelly (betting on the fixed score recommended by Kelly)
In recent years, Kelly has become a part of the mainstream investment theory, and claims to have made famous successful investors, including Warren Buffett and Bill Gross using Kelly method. William poundstone wrote an extensive account history of Kelly's gambling.
statement
For a simple bet, there are two outcomes, one involves losing the whole bet, and the other involves multiplying the amount won by the odds paid. Kelly's bet is:
f =[p(b+ 1)- 1]/b = BP-q/b
note:
F is the score of the current capital bet, that is, how much is bet;
B is the net odds received on the bet ("B to 1"), that is, you can win the bet of 1 USD (you can get back your bet of 1 USD on the bet);
P is the probability of winning;
Q is the probability of failure, which is 1? p；
For example, if a gambler's winning rate is 60% (p= 0.60, q = 0.40) and wins at the odds (b= 1) obtained by a gambler, then the gambler should bet 20% of his funds on every opportunity (f = 0.20) to maximize the long-term growth rate of funds.
If the gambler has zero side, that is, b = q/p, then the standard suggests that gamblers do not bet.
If the margin is negative (b= q/p), the formula gives a negative result, indicating that the gambler should accept the other side's bet.
For example, in the standard American roulette, when there are 18 red numbers and 20 non-red numbers (p = 18/38) in the roulette wheel, the bettor offers an even number of money prizes (b = 1) red. Kelly's bet is119, which means that the gambler should pay one-nineteenth of his Hong Qian. It's a pity that gambling is not allowed in casinos, so a Kelly gambler can't gamble.
At the top of the first part is a bet with an estimated net prize of $65,438+0, because the two results are that you will either win $ b with a probability of P or lose the bet of $65,438+0, that is, win $65,438+0 with a probability of Q, so:
F = expected net winnings/net winnings if
Even for money betting (that is, when b = 1), the first formula can be simplified as:
f=p-q
Since q = 1-p, this is further simplified as:
f=2p- 1
A more general question about investment decision-making is as follows:
The probability of success is p;
If successful, the value of your investment will increase by 1, which is1+b.
If it fails, (the probability is q= 1-P), the investment value drops to 1-a (note that the above description assumes that a is 1).
In this case, Kelly criterion becomes a relatively simple expression:
f=p/a-q/b
Note that this simplifies the original expression to a special case.
(f=p-q)，a=b= 1
Obviously, in order to decide to invest at least a small amount (F >; 0), which must be: pb> quality assurance.
This is obviously just a fact, that is, your expected profit must exceed your expected investment loss to be meaningful.
The general results explain why the use of loans will reduce the optimal proportion of investment. In this case, a> 1. Obviously, no matter how big the success probability p is, yes, if the value of A is large enough, the best score of investment is zero. Therefore, using too much margin is not a good investment strategy, no matter how good an investor you are.