凯利公式

 
在机率论中,凯利公式(也称凯利方程式)是一个用以使特定赌局中,拥有正期望值之重复行为长期增长率最大化的公式,由约翰·拉里·凯利於 1956 年在《贝尔系统技术期刊》中发表,可用以计算出每次游戏中应投注的资金比例。
简介
除可将长期增长率最大化外,此方程式不允许在任何赌局中,有失去全部现有资金的可能,因此有不存在破产疑虑的优点。方程式假设货币与赌局可无穷分割,而只要资金足够多,在实际应用上不成问题。

凯利公式

  凯利公式

凯利公式的最一般性陈述为,藉由寻找能最大化结果对数期望值的资本比例 f*,即可获得长期增长率的最大化。对於只有两种结果(输去所有注金,或者获得资金乘以特定赔率的彩金)的简单赌局而言,可由一般性陈述导出以下式子:

f*=(bp-q)/b
其中
f* 为现有资金应进行下次投注的比例;
b 为投注可得的赔率
p 为获胜率;
q 为落败率,即 1 – p;

凯利公式

  凯利公式

举例而言,若一赌博有 40% 的获胜率(p = 0.4,q = 0.6),而赌客在赢得赌局时,可获得二对一的赔率(b = 2),则赌客应在每次机会中下注现有资金的 10%(f* = 0.1),以最大化资金的长期增长率

凯利公式最初为 AT&T 贝尔实验室物理学家约翰·拉里·凯利根据同僚克劳德·艾尔伍德·夏农於长途电话线杂讯上的研究所建立。凯利说明夏农的资讯理论要如何应用於一名拥有内线消息的赌徒在赌马时的问题。赌徒希望决定最佳的赌金额,而他的内线消息不需完美(无杂讯),即可让他拥有有用的优势。凯利的公式随後被夏农的另一名同僚 爱德华·索普应用於二十一点和股票市场中。[1]

凯利公式(The Kelly Formula)的投资运用

凯利公式在投资中可作如下应用:
1、凯利公式不能代替选股,选股还是要按照巴菲特和费雪的方法。
2、凯利公式可以选时,即使是有投资价值的公式,也有高估和低估的时候,可以用凯利公式进行选时比较。
3、凯利公式适合非核心资产寻找短期投机机会。
4、凯利公式适合作为资产配置的考虑,对于资金管理比较有利,可以充分考虑机会成本。

凯利公式的盲点

凯利公式原本是为了协助规划电子比特流量设计,后来被引用于赌二十一点上去,麻烦就出在一个简单的事实,二十一点并非商品或交易。赌二十一点时,你可能会输的赌本只限于所放进去的筹码,而可能会赢的利润,也只限于赌注筹码的范围。但商品交易输赢程度是没得准的,会造成资产或输赢有很大的震幅

 

 

Kelly criterion

From Wikipedia, the free encyclopedia
  (Redirected from Kelly formula)

In probability theory, the Kelly criterionKelly strategyKelly formula, or Kelly bet, is a formula used to determine the optimal size of a series of bets. In most gambling scenarios, and some investing scenarios under some simplifying assumptions, the Kelly strategy will do better than any essentially different strategy in the long run. It was described by J. L. Kelly, Jr in 1956.[1] The practical use of the formula has been demonstrated.[2][3][4]

Although the Kelly strategy’s promise of doing better than any other strategy seems compelling, some economists have argued strenuously against it, mainly because an individual’s specific investing constraints may override the desire for optimal growth rate.[5] The conventional alternative is utility theory which says bets should be sized to maximize the expected utility of the outcome (to an individual with logarithmic utility, the Kelly bet maximizes utility, so there is no conflict). Even Kelly supporters usually argue for fractional Kelly (betting a fixed fraction of the amount recommended by Kelly) for a variety of practical reasons, such as wishing to reduce volatility, or protecting against non-deterministic errors in their advantage (edge) calculations.[6]

In recent years, Kelly has become a part of mainstream investment theory[7] and the claim has been made that well-known successful investors including Warren Buffett[8] and Bill Gross[9] use Kelly methods. William Poundstone wrote an extensive popular account of the history of Kelly betting.[5]

Statement[edit source | editbeta]

For simple bets with two outcomes, one involving losing the entire amount bet, and the other involving winning the bet amount multiplied by the payoff odds, the Kelly bet is:

 f^{*} = \frac{bp - q}{b} = \frac{p(b + 1) - 1}{b}, \!

where:

  • f* is the fraction of the current bankroll to wager;
  • b is the net odds received on the wager (“b to 1″); that is, you could win $b (plus the $1 wagered) for a $1 bet
  • p is the probability of winning;
  • q is the probability of losing, which is 1 − p.

As an example, if a gamble has a 60% chance of winning (p = 0.60, q = 0.40), but the gambler receives 1-to-1 odds on a winning bet (b = 1), then the gambler should bet 20% of the bankroll at each opportunity (f* = 0.20), in order to maximize the long-run growth rate of the bankroll.

If the gambler has zero edge, i.e. if b = q / p, then the criterion recommends the gambler bets nothing. If the edge is negative (b < q / p) the formula gives a negative result, indicating that the gambler should take the other side of the bet. For example, in standard American roulette, the bettor is offered an even money payoff (b = 1) on red, when there are 18 red numbers and 20 non-red numbers on the wheel (p = 18/38). The Kelly bet is -1/19, meaning the gambler should bet one-nineteenth of the bankroll that red will not come up. Unfortunately, the casino doesn’t allow betting against red, so a Kelly gambler could not bet.

The top of the first fraction is the expected net winnings from a $1 bet, since the two outcomes are that you either win $b with probability p, or lose the $1 wagered, i.e. win $-1, with probability q. Hence:

 f^{*} = \frac{\text{expected net winnings}}{\text{net winnings if you win}} \!

For even-money bets (i.e. when b = 1), the first formula can be simplified to:

 f^{*} = p - q . \!

Since q = 1-p, this simplifies further to

 f^{*} = 2p - 1 . \!

A more general problem relevant for investment decisions is the following:

1. The probability of success is p.

2. If you succeed, the value of your investment increases from 1 to 1+b.

3. If you fail (for which the probability is q=1-p) the value of your investment decreases from 1 to 1-a. (Note that the previous description above assumes that a is 1).

In this case, the Kelly criterion turns out to be the relatively simple expression

 f^{*} = p/a - q/b . \!

Note that this reduces to the original expression for the special case above (f^{*}=p-q) for b=a=1.

Clearly, in order to decide in favor of investing at least a small amount (f^{*}>0), you must have

 p b >  q a . \!

which obviously is nothing more than the fact that your expected profit must exceed the expected loss for the investment to make any sense.

The general result clarifies why leveraging (taking a loan to invest) decreases the optimal fraction to be invested , as in that case a>1. Obviously, no matter how large the probability of success, p, is, if a is sufficiently large, the optimal fraction to invest is zero. Thus using too much margin is not a good investment strategy, no matter how good an investor you are.

Proof[edit source | editbeta]

Heuristic proofs of the Kelly criterion are straightforward.[10] For a symbolic verification with Python and SymPy one would set the derivative y'(x) of the expected value of the logarithmic bankroll y(x) to 0 and solve for x:

>>> from sympy import *
>>> x,b,p = symbols('xbp')
>>> y = p*log(1+b*x) + (1-p)*log(1-x)
>>> solve(diff(y,x), x)
[-(1 - p - b*p)/b]

For a rigorous and general proof, see Kelly’s original paper[1] or some of the other references listed below. Some corrections have been published.[11]

We give the following non-rigorous argument for the case b = 1 (a 50:50 “even money” bet) to show the general idea and provide some insights.[1]

When b = 1, the Kelly bettor bets 2p – 1 times initial wealth, W, as shown above. If he wins, he has 2pW. If he loses, he has 2(1 – p)W. Suppose he makes N bets like this, and wins K of them. The order of the wins and losses doesn’t matter, he will have:

 2^Np^K(1-p)^{N-K}W \! .

Suppose another bettor bets a different amount, (2p – 1 + \Delta)W for some positive or negative \Delta. He will have (2p + \Delta)W after a win and [2(1 – p)- \Delta]W after a loss. After the same wins and losses as the Kelly bettor, he will have:

 (2p+\Delta)^K[2(1-p)-\Delta]^{N-K}W \!

Take the derivative of this with respect to \Delta and get:

 K(2p+\Delta)^{K-1}[2(1-p)-\Delta]^{N-K}W-(N-K)(2p+\Delta)^K[2(1-p)-\Delta]^{N-K-1}W\!

The turning point of the original function occurs when this derivative equals zero, which occurs at:

 K[2(1-p)-\Delta]=(N-K)(2p+\Delta) \!

which implies:

 \Delta=2(\frac{K}{N}-p) \!

but:

 \lim_{N \to +\infty}\frac{K}{N}=p \!

so in the long run, final wealth is maximized by setting \Delta to zero, which means following the Kelly strategy.

This illustrates that Kelly has both a deterministic and a stochastic component. If one knows K and N and wishes to pick a constant fraction of wealth to bet each time (otherwise one could cheat and, for example, bet zero after the Kth win knowing that the rest of the bets will lose), one will end up with the most money if one bets:

 \left(2\frac{K}{N}-1\right)W \!

each time. This is true whether N is small or large. The “long run” part of Kelly is necessary because K is not known in advance, just that as N gets large, Kwill approach pN. Someone who bets more than Kelly can do better if K > pN for a stretch; someone who bets less than Kelly can do better if K < pN for a stretch, but in the long run, Kelly always wins.

The heuristic proof for the general case proceeds as follows.[citation needed]

In a single trial, if you invest the fraction f of your capital, if your strategy succeeds, your capital at the end of the trial increases by the factor 1-f + f(1+b) = 1+fb, and, likewise, if the strategy fails, you end up having your capital decreased by the factor 1-fa. Thus at the end of Ntrials (with pN successes and qN failures ), the starting capital of $1 yields

C_N=(1+fb)^{pN}(1-fa)^{qN}.

Maximizing \log(C_N)/N, and consequently C_N, with respect to f leads to the desired result

f^{*}=p/a-q/b .

For a more detailed discussion of this formula for the general case, see http://www.bjmath.com/bjmath/thorp/ch2.pdf.

Reasons to bet less than Kelly[edit source | editbeta]

A natural assumption is that taking more risk increases the probability of both very good and very bad outcomes. One of the most important ideas in Kelly is that betting more than the Kelly amount decreases the probability of very good results, while still increasing the probability of very bad results. Since in reality we seldom know the precise probabilities and payoffs, and since overbetting is worse than underbetting, it makes sense to err on the side of caution and bet less than the Kelly amount.

Kelly assumes sequential bets that are independent (later work generalizes to bets that have sufficient independence). That may be a good model for some gambling games, but generally does not apply in investing and other forms of risk-taking.

The Kelly property appears “in the long run” (that is, it is an asymptotic property). To a person, it matters whether the property emerges over a small number or a large number of bets. It makes sense to consider not just the long run, but where losing a bet might leave one in the short and medium term as well. A related point is that Kelly assumes the only important thing is long-term wealth. Most people also care about the path to get there. Kelly betting leads to highly volatile short-term outcomes which many people find unpleasant, even if they believe they will do well in the end.

The criterion assumes you know the true value of p, the probability of the winning. The formula tells you to bet a positive amount if p is greater than 1/(b+1). In many situations you cannot be sure p is the true probability. For example if you are told there are just 100 tickets ($1 each) to a raffle, and the prize for winning is $110, then Kelly will tell you to bet a positive fraction of your bank. However, if the information of “100 tickets” was a lie or mis-estimate, and if the true number of tickets was 120, then any bet needs to be avoided. Your optimal investement strategy will need to consider the statistical distribution for your estimate for p.

Bernoulli[edit source | editbeta]

In a 1738 article, Daniel Bernoulli suggested that when one has a choice of bets or investments that one should choose that with the highest geometric mean of outcomes. This is mathematically equivalent to the Kelly criterion[citation needed], although the motivation is entirely different (Bernoulli wanted to resolve theSt. Petersburg paradox). The Bernoulli article was not translated into English until 1956,[12] but the work was well-known among mathematicians and economists.

Many horses[edit source | editbeta]

Kelly’s criterion may be generalized [13] on gambling on many mutually exclusive outcomes, like in horse races. Suppose there are several mutually exclusive outcomes. The probability that the k-th horse wins the race is p_k, the total amount of bets placed on k-th horse is B_k, and

\beta_k=\frac{B_k}{\sum_i B_i}=\frac{1}{1+Q_k} ,

where Q_k are the pay-off odds. D=1-tt, is the dividend rate where tt is the track take or tax, \frac{D}{\beta_k} is the revenue rate after deduction of the track take when k-th horse wins. The fraction of the bettor’s funds to bet on k-th horse is f_k. Kelly’s criterion for gambling with multiple mutually exclusive outcomes gives an algorithm for finding the optimal set S^o of outcomes on which it is reasonable to bet and it gives explicit formula for finding the optimal fractions f^o_k of bettor’s wealth to be bet on the outcomes included in the optimal set S^o. The algorithm for the optimal set of outcomes consists of four steps.[13]

Step 1 Calculate the expected revenue rate for all possible (or only for several of the most promising) outcomes: er_k=\frac{D}{\beta_k}p_k=D(1+Q_k)p_k.

Step 2 Reorder the outcomes so that the new sequence er_k is non-increasing. Thus er_1 will be the best bet.

Step 3 Set  S = \varnothing  (the empty set), k = 1R(S)=1. Thus the best bet er_k = er_1 will be considered first.

Step 4 Repeat:

If er_k=\frac{D}{\beta_k}p_k > R(S) then insert k-th outcome into the set: S = S \cup \{k\}, recalculate R(S) according to the formula: R(S)=\frac{1-\sum_{i \in S}{p_i}}{1-\sum_{i \in S } \frac{\beta_i}{D}} and then set k = k+1 ,

Else set S^o=S and then stop the repetition.

If the optimal set S^o is empty then do not bet at all. If the set S^o of optimal outcomes is not empty then the optimal fraction f^o_k to bet on k-th outcome may be calculated from this formula: f^o_k=\frac{er_k - R(S^o)}{\frac{D}{\beta_k}}=p_k-\frac{R(S^o)}{\frac{D}{\beta_k}}.

One may prove[13] that

R(S^o)=1-\sum_{i \in S^o}{f^o_i}

where the right hand-side is the reserve rate[clarification needed]. Therefore the requirement er_k=\frac{D}{\beta_k}p_k > R(S) may be interpreted[13] as follows: k-th outcome is included in the set S^o of optimal outcomes if and only if its expected revenue rate is greater than the reserve rate. The formula for the optimal fraction f^o_k may be interpreted as the excess of the expected revenue rate of k-th horse over the reserve rate divided by the revenue after deduction of the track take when k-th horse wins or as the excess of the probability of k-th horse winning over the reserve rate divided by revenue after deduction of the track take when k-th horse wins. The binary growth exponent is

G^o=\sum_{i \in S}{p_i\log_2{(er_i)}}+(1-\sum_{i \in S}{p_i})\log_2{(R(S^o))} ,

and the doubling time is

T_d=\frac{1}{G^o}.

This method of selection of optimal bets may be applied also when probabilities p_k are known only for several most promising outcomes, while the remaining outcomes have no chance to win. In this case it must be that \sum_i{p_i} < 1 and \sum_i{\beta_i} < 1.

Application to the stock market[edit source | editbeta]

Consider a market with n correlated stocks S_k with stochastic returns r_kk= 1,...,n and a riskless bond with return r. An investor puts a fraction u_k of his capital in S_k and the rest is invested in bond. Without loss of generality assume that investor’s starting capital is equal to 1. According to Kelly criterion one should maximize \mathbb{E}\left[ \ln\left((1 + r) + \sum\limits_{k=1}^n  u_k(r_k -r) \right) \right]
Expanding it to the Taylor series around \vec{u_0} = (0, \ldots ,0) we obtain
\mathbb{E} \left[ \ln(1+r) + \sum\limits_{k=1}^{n} \frac{u_k(r_k - r)}{1+r} -<br />
\frac{1}{2}\sum\limits_{k=1}^{n}\sum\limits_{j=1}^{n} u_k u_j \frac{(r_k<br />
-r)(r_j - r)}{(1+r)^2} \right]
Thus we reduce the optimization problem to the Quadratic programming and the unconstrained solution is <br />
\vec{u^{\star}} = (1+r) (  \widehat{\Sigma} )^{-1} ( \widehat{\vec{r}}  )<br />
where \widehat{\vec{r}} and \widehat{\Sigma} are the vector of means and the matrix of second mixed noncentral moments of the excess returns.[14] There are also numerical algorithms for the fractional Kelly strategies and for the optimal solution under no leverage and no short selling constraints.